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By Goro Shimura

ISBN-10: 0691016569

ISBN-13: 9780691016566

I regard the publication as a invaluable gate to the information during which "Fermat's final theorem" has been concluded. accordingly for any mathematician who wish to grasp in algebraic geometry, quantity thought or any alike topic it really is an crucial source of first look.

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Additional resources for Abelian Varieties with Complex Multiplication and Modular Functions

Example text

N We take Haar measure dµ =I. fl Ol{J. 1 = for the measure on 11{ = Torn. z1, ... , e 27rik t n zn) or St(z) = (

E ), -ao < t < ao is the trajectory of zero of the torus Torn looked at as a group. parameter subgroup of the torus Torn, It forms a one- and hence {st} is a group of trans- lations on the torus Torn in the direction of the subgroup {a(t)}. The flow {st} is often called conditionally periodic, and ki-its frequencies. Theorem. •• , kn are linearly independent over the field of rational numbers. 49 Proof. First we remark that in the case of a flow on the torus, as in the discrete case, the trajectory of all points are simultaneously everywhere dense on the torus.

The problem of the distribution of the fractional parts of polynomials is related to the calculation of the number of solutions to diophantine equations in the algebraic theory of numbers. The interested reader can find further information in H. Furstenberg, Strict ergodicity and transformation of the torus. Amer. J. of Math. , 1961, v. 83, NY, 573-601, and A. G. Postnikov , Ergodic problems of the theory of congruences and Diophant approximations. v. 8Z, 1966. Proceedings of Steklov Institute, 39 Lecture 5 A SECOND PROOF OF THE ERGODICITY OF THE ROTATION OF A CIRCLE AND PERMUTATIONS We will give a new proof of the ergodicity of the rotation of a circle through an irrational angle without using the expansion of a function into its Fourier series.

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Abelian Varieties with Complex Multiplication and Modular Functions by Goro Shimura

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